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Question
Find the area of the shaded region in the figure as shown, in which DPQS is an equilateral triangle and ∠PQR = 90°.
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Solution
In right-angled ΔPPQR,
PR2 = PQ2 + RQ2
⇒ RQ2 = PR2 - PQ2
= 202 - 122
= 400 - 144
= 256
⇒ RQ = 16cm
Area of ΔPQR
= `(1)/(2) xx "PQ" xx "RQ"`
= `(1)/(2) xx 12 xx 16`
= 96cm2
Area of equilateral ΔPQS
= `sqrt(3)/(4) xx ("side")^2`
= `sqrt(3)/(4) xx 12 xx 12`
= `36sqrt(3)"cm"^2`
Now,
area of shaded region
= Area of ΔPQR - Area of equilateral ΔPQS
= 96 - `36sqrt(3)`
= `12(8 - 3sqrt(3))"cm"^2`.
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